A251599 Centers of rows of the triangular array formed by the natural numbers.

1, 2, 3, 5, 8, 9, 13, 18, 19, 25, 32, 33, 41, 50, 51, 61, 72, 73, 85, 98, 99, 113, 128, 129, 145, 162, 163, 181, 200, 201, 221, 242, 243, 265, 288, 289, 313, 338, 339, 365, 392, 393, 421, 450, 451, 481, 512, 513, 545, 578, 579, 613, 648, 649, 685, 722, 723

Offset: 1

Author

Dave Durgin, Dec 05 2014

Keywords

• nonn

Comments

Forms a cascade of 3-number triangles down the center of the triangle array. Related to A000124 (left/west bank of same triangular array), A000217 (right/east bank) and A001844 (center column).

Sums of the mentioned cascading triangles: a(3*n-2) + a(3*n-1) + a(3*n) = A058331(n) + A001105(n) + A001844(n-1) = 2*A056106(n) = 2*(3*n^2-n+1). - Reinhard Zumkeller, Dec 13 2014

Union of A080827 and A000982. - David James Sycamore, Aug 09 2018

Examples

			First ten terms (1,2,3,5,8,9,13,18,19,25) may be read down the center of the triangular formation:
               1
             2   3
           4   5   6
         7   8   9  10
      11  12  13  14  15
    16  17  18  19  20  21
  22  23  24  25  26  27  28
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• David James Sycamore, A080827 & A000982
• Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Crossrefs

Cf. A000124, A000217, A001844.

Cf. A092942 (first differences).

Cf. A001105, A056106, A058331.

Cf. A080827, A000982.

Programs

Haskell

a251599 n = a251599_list !! (n-1)
a251599_list = f 0 $ g 1 [1..] where
   f i (us:vs:wss) = [head $ drop i us] ++ (take 2 $ drop i vs) ++
                     f (i + 1) wss
   g k zs = ys : g (k + 1) xs where (ys,xs) = splitAt k zs
-- Reinhard Zumkeller, Dec 12 2014

Maple

a:= n-> (m-> 2*(m+1)^2-[2*m+1, 0, -1][1+r])(iquo(n-1, 3, 'r')):
seq(a(n), n=1..100);  # Alois P. Heinz, Dec 10 2014

Mathematica

LinearRecurrence[{1, 0, 2, -2, 0, -1, 1}, {1, 2, 3, 5, 8, 9, 13}, 60] (* Jean-François Alcover, Jan 09 2016 *)

PARI

Vec(-x*(x^2+1)*(x^4-x^3+x+1)/((x^2+x+1)^2*(x-1)^3) + O(x^80)) \\ Michel Marcus, Jan 09 2016

Formula

Terms for n=1 (mod 3): 2m^2+2m+1, for n=2 (mod 3): 2m^2+4m+2, for n=0 (mod 3): 2m^2+4m+3, where m = floor((n-1)/3).

G.f.: -x*(x^2+1)*(x^4-x^3+x+1)/((x^2+x+1)^2*(x-1)^3). - Alois P. Heinz, Dec 10 2014